To factor the expression [tex]\( 12 x^3 y^5 - 20 x^5 y^2 z \)[/tex], we follow these steps:
1. Identify the greatest common factor (GCF):
We start by finding the greatest common factor of the terms in the expression.
- For the coefficients 12 and 20, the GCF is 4.
- For the variable [tex]\( x \)[/tex], the lowest power present in both terms is [tex]\( x^3 \)[/tex], so the GCF is [tex]\( x^3 \)[/tex].
- For the variable [tex]\( y \)[/tex], the lowest power present in both terms is [tex]\( y^2 \)[/tex], so the GCF is [tex]\( y^2 \)[/tex].
Combining these, the overall GCF of the expression is [tex]\( 4 x^3 y^2 \)[/tex].
2. Factor out the GCF:
We now factor out [tex]\( 4 x^3 y^2 \)[/tex] from each term in the expression.
- For the first term [tex]\( 12 x^3 y^5 \)[/tex]:
[tex]\[
\frac{12 x^3 y^5}{4 x^3 y^2} = 3 y^3
\][/tex]
- For the second term [tex]\( 20 x^5 y^2 z \)[/tex]:
[tex]\[
\frac{20 x^5 y^2 z}{4 x^3 y^2} = 5 x^2 z
\][/tex]
So, factoring [tex]\( 4 x^3 y^2 \)[/tex] out from the given expression, we get:
[tex]\[
12 x^3 y^5 - 20 x^5 y^2 z = 4 x^3 y^2 (3 y^3) - 4 x^3 y^2 (5 x^2 z)
\][/tex]
3. Express the factored form:
Combining the factored terms, the expression becomes:
[tex]\[
4 x^3 y^2 (3 y^3 - 5 x^2 z)
\][/tex]
Therefore, the factored form of [tex]\( 12 x^3 y^5 - 20 x^5 y^2 z \)[/tex] is:
[tex]\[
4 x^3 y^2 (3 y^3 - 5 x^2 z)
\][/tex]
